The BGLIMM Procedure

specifies the prior to be a half-Cauchy distribution. The half-Cauchy prior is applied only to the diagonal terms (variances) of the matrix. The off-diagonal terms of the matrix are assumed to have a flat prior.

You can specify the scale parameter a of the half-Cauchy distribution. The scale parameter a must be positive. By default, SCALE=25.

specifies the prior to be a half-normal distribution. The half-normal prior is applied only to the diagonal terms (variances) of the matrix. The off-diagonal terms of the matrix are assumed to have a flat prior.

You can specify the variance a of the half-normal distribution. The variance a must be positive. By default, VAR=25.

specifies an inverse gamma prior, , with the density for each diagonal term of the matrix. It is the default prior for the covariance types UN(1), VC, and TOEP(1).

You can set the parameters of the inverse gamma distribution by specifying one or both of the following options, separated by a comma or a space:

specifies an inverse Wishart prior, , for the matrix of the random effects. The hyperparameters a and are the degrees of freedom and scale matrix of an inverse Wishart distribution, respectively. The inverse Wishart prior is the default prior for the UN covariance type. The following examples show several ways to specify an inverse Wishart prior distribution:

covprior=iwishart covprior=iwishart(scale=25) covprior=iwishart(df=6, diagonal=(1 4 9))

You can set the parameters of the inverse Wishart distribution by specifying one or more of the following options, separated by a comma or a space:

specifies a scaled inverse Wishart prior for the matrix of the random effects.

You can set the parameters of the scaled inverse Wishart distribution by specifying one or more of the following options, separated by a comma or a space:

specifies a uniform prior, , for the matrix of the random effects. The uniform prior is applied to standard deviations (the square root of the diagonal terms) of the matrix.

You can set the lower and upper bounds of the uniform distribution by specifying one or both of the following options, separated by a comma or a space:

specifies the target acceptance rate during the tuning process of the NUTS algorithm. Increasing the value can often improve mixing, but it can also significantly slow down the sampling. By default, DELTA=0.6.

specifies the maximum height of the NUTS algorithm tree. The taller the tree, the more gradient evaluations per iteration the procedure calculates. The number of evaluations is . Usually, the height of a tree should be no more than 10 during the sampling stage, but it can go higher during the tuning stage. A larger number indicates that the algorithm is having difficulty converging. By default, MAXHEIGHT=10.

specifies the number of tuning iterations for the NUTS algorithm to use. By default, NTU=1000.

specifies the initial step size in the NUTS algorithm. By default, STEPSIZE=0.1.

specifies a first-order antedependence structure (Kenward 1987; Patel 1991) parameterized in terms of variances and correlation parameters. If t ordered random variables have a first-order antedependence structure, then each , , is independent of all other , given . This Markovian structure is characterized by its inverse variance matrix, which is tridiagonal. Parameterizing an ANTE(1) structure for a random vector of size t requires 2t – 1 parameters: variances and t – 1 correlation parameters . The covariances among random variables and are then constructed as

PROC BGLIMM constrains the correlation parameters to satisfy . For variable-order antedependence models see Macchiavelli and Arnold (1994).

specifies a first-order autoregressive structure,

The values and are derived for the ith and jth observations, respectively, and are not necessarily the observation numbers. For example, in the following statements, the values correspond to the class levels for the Time effect of the ith and jth observation within a particular subject:

proc bglimm;
   class time patient;
   model y = x x*x;
   random time / sub=patient type=ar(1);
run;

PROC BGLIMM imposes the constraint for stationarity.

specifies a heterogeneous first-order autoregressive structure,

where . This covariance structure has the same correlation pattern as the TYPE=AR(1) structure, but the variances are allowed to differ.

specifies the first-order autoregressive moving average structure,

Here, is the autoregressive parameter, models a moving average component, and is a scale parameter. In the notation of Fuller (1976, p. 68), and

The example in Table 11 and imply that

where and . PROC BGLIMM imposes the constraints and for stationarity, although for some values of and in this region, the resulting covariance matrix is not positive definite.

specifies the compound symmetry structure, which has constant variance and constant covariance

The compound symmetry structure arises naturally with nested random effects, such as when subsampling error is nested within experimental error. Hierarchical random assignments or selections, such as subsampling or split-plot designs, give rise to compound symmetric covariance structures. This implies exchangeability of the observations in the subunit, leading to constant correlations between the observations. Compound symmetry structures are thus usually not appropriate for processes in which correlations decline according to some metric, such as spatial and temporal processes.

specifies the heterogeneous compound symmetry structure, which is an equicorrelation structure but allows for different variances,

specifies the factor-analytic structure with one factor (Jennrich and Schluchter 1986). This structure is of the form , where is a vector and is a diagonal matrix with t different parameters.

specifies a covariance structure that satisfies the general Huynh-Feldt condition (Huynh and Feldt 1970). For a random vector that has t elements, this structure has positive parameters and covariances

A covariance matrix generally satisfies the Huynh-Feldt condition if it can be written as . The preceding parameterization chooses . Several simpler covariance structures give rise to covariance matrices that also satisfy the Huynh-Feldt condition. For example, TYPE=CS, VC, and UN(1) are nested within TYPE=HF. Note also that the HF structure is nested within an unstructured covariance matrix.

models a Toeplitz covariance structure. This structure can be viewed as an autoregressive structure whose order is equal to the dimension of the matrix,

specifies a banded Toeplitz structure,

This can be viewed as a moving average structure whose order is equal to q – 1. The specification TYPE=TOEP(1) is the same as , and it can be useful for specifying the same variance component for several effects.

models a Toeplitz covariance structure. The correlations of this structure are banded as in the TOEP or TOEP(q) structures, but the variances are allowed to vary:

The correlation parameters satisfy . If you specify the optional value q, the correlation parameters with are set to 0, creating a banded correlation structure. The specification TYPE=TOEPH(1) results in a diagonal covariance matrix with heterogeneous variances.

specifies a completely general (unstructured) covariance matrix that is parameterized directly in terms of variances and covariances,

The variances are constrained to be nonnegative, and the covariances are unconstrained. This structure is constrained to be nonnegative definite. If you specify the order parameter q, then PROC BGLIMM estimates only the first q bands of the matrix, setting elements in all higher bands equal to 0.

specifies standard variance components and is the default structure for both G-side and R-side covariance structures. In a G-side covariance structure, a distinct variance component is assigned to each effect. In an R-side structure, TYPE=VC is usually used only to add overdispersion effects or, with the GROUP= option, to specify a heterogeneous variance model.